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Mirrors > Home > MPE Home > Th. List > 0cld | Structured version Visualization version GIF version |
Description: The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
0cld | ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4384 | . . 3 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽 | |
2 | 1 | topopn 22928 | . 2 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ ∅) ∈ 𝐽) |
3 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
4 | eqid 2735 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | iscld2 23052 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
6 | 3, 5 | mpan2 691 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
7 | 2, 6 | mpbird 257 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ‘cfv 6563 Topctop 22915 Clsdccld 23040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-top 22916 df-cld 23043 |
This theorem is referenced by: cls0 23104 indiscld 23115 iscldtop 23119 iccordt 23238 isconn2 23438 tgptsmscld 24175 mblfinlem2 37645 mblfinlem3 37646 ismblfin 37648 |
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